Universal finite-size scaling for the $|\varphi|^4$ model in 4 dimensions
and higher
We discuss recent work establishing precise finite-size scaling of
statistical physics models at and above the upper critical dimension. We
prove that, close enough to the infinite volume critical point, in a
volume-dependent critical window, models defined in a large box behave
critically.
For the $|\varphi|^4$ model in 4 dimensions and above, we identify: the
width of the critical window, the limiting distribution of the total
field and the role of boundary conditions in our results. We also review
to what extent we believe our results are universal, and list a few open
problems.
This is based on joint work with Jiwoon Park and Gordon Slade.